Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws
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Date
2019-06
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Abstract
Using the theoretical framework of algebraic flux correction and invariant domain
preserving schemes, we introduce a monolithic approach to convex limiting
in continuous finite element schemes for linear advection equations, nonlinear
scalar conservation laws, and hyperbolic systems. In contrast to fluxcorrected
transport (FCT) algorithms that apply limited antidiffusive corrections
to bound-preserving low-order solutions, our new limiting strategy exploits
the fact that these solutions can be expressed as convex combinations
of bar states belonging to a convex invariant set of physically admissible solutions.
Each antidiffusive flux is limited in a way which guarantees that the
associated bar state remains in the invariant set and preserves appropriate local
bounds. There is no free parameter and no need for limit fluxes associated with
the consistent mass matrix of time derivative term separately. Moreover, the
steady-state limit of the nonlinear discrete problem is well defined and independent
of the pseudo-time step. In the case study for the Euler equations, the
components of the bar states are constrained sequentially to satisfy local maximum
principles for the density, velocity, and specific total energy in addition
to positivity preservation for the density and pressure. The results of numerical
experiments for standard test problems illustrate the ability of built-in convex
limiters to resolve steep fronts in a sharp and nonoscillatory manner.
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Keywords
hyperbolic conservation laws, positivity preservation, invariant domain, finite elements, algebraic flux correction, convex limiting